AMS Fall Western Sectional Meeting

Let me begin with a very general statement about the meeting. It was simply amazing! I met many amazing people – Dr. Marcy Robertson, Dr. Julie Bergner, Dr. Aaron Mazel-Gee, Dr. Matthew Pancia, Dr. Eric Peterson and Dr. David Carchedi. They were all very nice people, and it was very fun meeting them. Here are the highlights, for me, of the meeting.

First, my own presentation at 10:30 in the morning. My own presentation went very well; here is the final version of the slides that I presented: AMS Fall Western Sectional Meeting.

Then I visited Dr. Kiran Kedlaya’s lecture, A brief history of perfectoid spaces. Essentially, perfectoid spaces are things that connect stuff in characteristic 0 and characteristic p. He never really presented the actual precise definition of a perfectoid space; he said that the precise definition takes one away from the actual intended use of the objects (whose purpose is to connect stuff in characteristic 0 and characteristic p). I then ate lunch at Subway on the SFSU campus.

Following that tasty lunch, I went on to visit Dr. Marcy Robertson’s talk, which was at 3:00 pm. Since I didn’t know where everyone was, and my parents had taken my brother out to the mall nearby (he was understandably bored), I sat in the room two hours early. Just before I was about to leave out of boredom, Dr. Robertson came in.

When I’d first met her in UCLA, she’d given my some stuff to read on symmetric spectra by Hovey, Shipley, et. al. She told me some interesting stuff about DGAs and DG-categories (DG-categories with one object are DGAs!), and it was very fun! She then told me about how my research on the K-theory of modules over infinity operads could actually provide some input into the theory of derived categories.

By then, people started coming in, and I met Dr. Julie Bergner, who was a very nice person. Dr. Robertson’s talk was on Morita theory and categorification. I’ve taken some notes, and plan to type them in sometime soon. Dr. Eric Peterson’s talk was on determinantal K-theory and some applications; his typed in notes are here.

After the talks, there was really nothing much to do, so I went out, wandered about like an idiot, and came back into the room. Dr. Robertson and Dr. Bergner were leaving then, and I went into the room and introduced myself to Dr. Aaron Mazel-Gee, Dr. Matthew Pancia, Dr. Eric Peterson and Dr. David Carchedi. They were very nice people, and I talked to them about my research. They offered to put me in contact with Dr. Barwick and Dr. Lurie, and they also gave me their emails. They were very nice people, and I hope to see them sometime soon!

There was the Einstein Public Lecture, of course, by Dr. Simons – famous for Chern-Simons theory. I didn’t attend the reception, but I did attend the lecture.

I didn’t stay for the second day, as I have school tomorrow, and the drive was very very long – 7 hours! I hope to post more regularly, but until then, goodbye!

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Congratulations to the 2014 Fields Medalists!

Congratulations to the 2014 Fields Medalists (http://www.mathunion.org/general/prizes/2014)!

Here’s a brief bio about them, from the Simons Foundation, for those interested:

  • Artur Avila – http://www.simonsfoundation.org/quanta/20140812-a-brazilian-wunderkind-who-calms-chaos/
  • Manjul Bhargava – http://www.simonsfoundation.org/quanta/20140812-the-musical-magical-number-theorist/
  • Martin Hairer – http://www.simonsfoundation.org/quanta/20140808-in-mathematical-noise-one-who-heard-music/
  • Maryam Mirzakhani – http://www.simonsfoundation.org/quanta/20140812-a-tenacious-explorer-of-abstract-surfaces/

Some intel (http://www.math.columbia.edu/~woit/wordpress/?p=7085&cpage=1) informs me that “the Fields Medal Committee was: Daubechies, Ambrosio, Eisenbud, Fukaya, Ghys, Dick Gross, Kirwan, Kollar, Kontsevich, Struwe, Zeitouni and Günter Ziegler”.

 

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The Riemann-Hurwitz Formula

This post is going to be one of those “occasional posts” that I talked about in the About page, which, as the reader has seen, is about the Riemann-Hurwitz formula.

I had already learnt about this formula a few years ago, and again a few weeks ago while perusing Hartshorne’s Algebraic Geometry, and came across it yet again while reading (or should I say glancing through) Diamond and Shurman’s A First Course in Modular Forms (which I call [DS] from now).love category theory, and am not an expert at number theory – this book seemed like an interesting book, and so I borrowed it from my local library. (The only parts I fully understood were the sections on topology, charts, and Riemann surfaces.)

Now to actual math. In [DS], section 3.1, one has the Riemann-Hurwitz formula. I’m going to try to explain this from a number-theorist point of view, so please feel free to point out any mistakes.

Let X and Y be two compact Riemann surfaces, with a nonconstant holomorphic map f:X\to Y. The first theorem is:

Theorem: f:X\to Y is surjective.

Proof: The image f(X) must be closed and open, thus Y-f(X) is open, since compact sets are closed in Hausdorff spaces. Thus, Y is disconnected, hence a contradiction. Q.E.D.

f has a number associated to it, known as the degree, a number c\in \mathbb{Z}^+ such that |f^{-1}(y)|=d for all but finitely many y\in Y, which is defined by the following construction. Let e_x denote the “multiplicity with which f takes 0 to 0 as a map in local coordinates, making f an $latex e_x$-to-1 map about x” ([DS]). Then, the degree is the positive integer such that \sum_{x\in f^{-1}(y)}e_x=c for all y\in Y. (For a proof of the existence of c, see [DS, section 3.1].)

Let g_X and g_Y denote the genera of $latex X$ and Y respectively (“genera” sounds weird, at least to me – but so does “genuses” – I guess “genii” would make sense, but that sounds a lot like the plural of “genie”…). The Riemann-Hurwitz formula is as follows:

Theorem (Riemann-Hurwitz Formula): 

2g_X-2=\sum_{x\in X}(e_x-1)+c(2g_Y-2)

Proof Sketch: There’s a very interesting proof (sketch) of this theorem, as [DS] gives. First, define \mathcal{C}:=\{x\in X| e_x>1\}. Triangulate Y using V_Y vertices including all points of f(\mathcal{C}), F_Y faces and E_Y edges. Under f^{-1}, we may lift this to a triangulation of X with E_X=cE_Y edges, F_X=cF_Y faces, and by ramification, V_X=cV_Y-\sum_{x\in X}(e_x-1) vertices. Since 2-2g_X=F_X-E_X+V_X and 2-2g_Y=F_Y-E_Y+V_Y, the Riemann-Hurwitz formula follows.

A very detailed proof, which is in essence what I did when I expanded out the proof sketch in [DS], is here.

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About me (and the blog)

I am Sanath Devalapurkar, a high-school student. This blog will be a website of mine, where I will post some of my ideas on algebraic topology with an occasional sprinkle of a few other topics from mathematics which I find interesting. I’ll use this blog as a medium to display some of my ideas, so feel free to comment.

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